JNTU
JNTU B.Tech I MATHEMATICS-I Previous Question Paper (June - 2022 - R18)
Answer any five questions. All questions carry equal marks.
Main
Apply rank test to find whether the following system has any solution other than x = y = z = w = 0, x + 2y + 3z + 4w = 0, 8x + 5y + z + 4w = 0, 5x + 6y + 8z + w = 0, 8x + 3y + 7z + 2w = 0.
Answer
The given system of homogeneous linear equations is AX = 0, where the coefficient matrix A is:
We perform row reduction to find the rank of A:
- R2 -> R2 - 8R1: [[1, 2, 3, 4], [0, -11, -23, -28], [5, 6, 8, 1], [8, 3, 7, 2]]
- R3 -> R3 - 5R1: [[1, 2, 3, 4], [0, -11, -23, -28], [0, -4, -7, -19], [8, 3, 7, 2]]
- R4 -> R4 - 8R1: [[1, 2, 3, 4], [0, -11, -23, -28], [0, -4, -7, -19], [0, -13, -17, -30]]
- R2 -> R2 / -11: [[1, 2, 3, 4], [0, 1, 23/11, 28/11], [0, -4, -7, -19], [0, -13, -17, -30]]
- R3 -> R3 + 4R2 and R4 -> R4 + 13R2: The resulting matrix has 4 non-zero rows, meaning rank(A) = 4.
Since the number of unknowns n = 4 and rank(A) = 4, the system has only the trivial solution x = y = z = w = 0. There are no non-zero solutions.
By using Gauss’s elimination method solve 5x-y-2z=142, x-3y-z=-30, 2x-y-3z = -5.
Answer
The augmented matrix [A|B] is:
Applying Gauss elimination:
- Swap R1 and R2: [[1, -3, -1, -30], [5, -1, -2, 142], [2, -1, -3, -5]]
- R2 -> R2 - 5R1 and R3 -> R3 - 2R1: [[1, -3, -1, -30], [0, 14, 3, 292], [0, 5, -1, 55]]
- R3 -> R3 - (5/14)R2: [[1, -3, -1, -30], [0, 14, 3, 292], [0, 0, -29/14, -450/14]]
- From R3: -29z = -450, so z = 450/29 ≈ 15.52.
- From R2: 14y + 3(450/29) = 292, so 14y = 292 - 1350/29 = 7118/29, y = 508.4/29 ≈ 17.53.
- From R1: x - 3(17.53) - 15.52 = -30, x = 23.07.
The solution is x ≈ 23.07, y ≈ 17.53, z ≈ 15.52.
Find the characteristic polynomial of the matrix A = [[3, 1, 1], [1, 5, -1], [1, -1, 3]] and verify Cayley-Hamilton theorem for this matrix. Hence find A-1, if exist.
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The matrix A = [[a, h], [h, b]] is transformed to the diagonal form D = T-1AT, where T = [[cosθ, sinθ], [-sinθ, cosθ]]. Find the value of θ which gives this diagonal transformation.
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Prove that the following series for conditional convergence: i) Σ (-1)n-1/np ii) Σ (-1)n-1 (1.3.5...(2n-1)) / (2.4.6...(2n)).
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Prove that the series sin x - (sin 2x)/2 + (sin 3x)/3 - ... converges absolutely.
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Express the log(cos x) as Taylor series about x = π/3.
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Find the volume of solid generated by revolving the ellipse x2/a2 + y2/b2 = 1 (b < a) rotated about minor axis.
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If u = xf(x, y) + yg(x, y), show that ∂2u/∂x2 - 2∂2u/∂x∂y + ∂2u/∂y2 = 0.
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Determine whether u and v are functionally dependent, where u and v are defined by u = sin x + cos y and v = cos x + sin y.
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Find the greatest value of u = xyz, if x, y and z are positive real numbers for which 4x + 2y + z = 12.
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A function f(x, y) is written in terms of new variables u = ex cos y and v = ex sin y. Show that ∂f/∂x = ∂f/∂u + ∂f/∂v and ∂f/∂y = -∂f/∂u + ∂f/∂v, and so deduce that.
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By using techniques involving the Gamma function, find the value of ∫ xn e-x dx from 0 to infinity.
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By using techniques involving the Beta function, find the exact value of ∫ sin2 x cos4 x dx from 0 to π/4.
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